Matrix Regularization of Symplectic and Conformally Invariant Theories

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Matrix Regularization of Symplectic and Conformally Invariant Theories JJ S. G. Rajeev

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University of Rochester

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email: [email protected]

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Branes Workshop at Argonne National Laboratories, Oct 2003

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Symplectic and Conformal Transformations 2/24

A metric gij on a 2-manifold Σ defines: (i) an area element (invertible 2-form) ωij = (ii) and a complex structure Jji =

√

gij

√1 g ik kj . g

These contain complementary pieces of information about the metric, the first invariant under area preserving (symplectic) diffeomorphisms

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while the second is invariant under conformal transformations.

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Symplectic Field Theories 3/24

There are thus two classes of field theories whose continuum limits are invariant under these transformations. An example of a symplectic field theory is two dimensional Yang–Mills R theory, S = Σ tr F (A) ∗ F (A). This is exactly solvable. A more intricate example is the classical field theory of incompressible fluid flow in two dimensions, since it can be viewed as the geodesic equations on the group of area preserving diffeomorphisms (Arnold). ∂u + u · ∇u = −∇p, ∂t

∇ · u = 0.

(1)

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It is best to eliminate pressure by taking the curl of this equation, writing it in terms of vorticity ω = ∇ × u: Z ∂ω = ∂aωab∂b G(x, y)ω(y)d2y. ∂t

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Here, G(x, y) is the Green’s function of the Laplace operator: vorticity determines velocity through Z ua(x) =

abω(y)∂bG(x, y)d2y

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The analogy of this to the equations of a rigid body dL = L × (I −1L) dt

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could not have escaped Euler. Vorticity is analogous to angular momentum; the Laplace operator is analogous to moment of inertia I. 5/24


Hamiltonian Formalism of Euler Equations 6/24

The analogue of the angular momentum Lie algebra is the algebra of symplectic transformations on the plane: [f1, f2] = ab∂af1∂bf2.

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Every function f : R2 → R corresponds to an observable of two dimenR sional hydrodynamics, ωf = f (x)ω(x)d2x. The above Lie bracket of

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functions gives the Poisson bracket for vorticity:

I {ω(x), ω(y)} = ab∂bω(x)∂aδ(x − y).

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Euler equations follow from these if we postulate the hamiltonian to be the total energy of the fluid, Z Z 1 1 H= u2d2x = G(x, y)ω(x)ω(y)d2xd2y. 2 2

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The rigid body equations are the equations for a geodesic in the rotation group, with respect to the metric defined by the moment of inertia. In the same way, the Euler equations of hydrodynamics describe geodesics in the group of symplectic diffeomorphisms with respect to

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the anisotropic metric defined by the kinetic energy. R The quantities Qk = ω k (x)d2x are conserved for any k = 1, 2 · · · :

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these are the Casimir invariants. In spite of the infinite number of

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conservation laws, two dimensional fluid flow is chaotic.

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The Stochastic Navier-Stokes Equation 8/24

A chaotic system is sensitive to small changes in initial conditions or its environment. We can model this by adding a Gaussian noise to the equations. Any time we have noise, we must also have dissipationotherwise the system will heat up to infinite energy. The dissipation of hydrodynamics is modelled well by viscosity ( not necessarily the

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molecular viscosity).

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The Navier-Stokes equations with a random force field1 Du = −∇p + ν∆u + f Dt

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would model this situation nicely. Taking its curl to eliminate pressure

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again, we get ∂ω = ∂aωab∂b ∂t

Z

G(x, y)ω(y)d2y − ν∆ω + η.

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JJ The dissipative term can be viewed as the gradient of energy H = 1 −1 2 (ω, ∆ ω),

with respect to the contravariant metric tensor in the vector

space of vorticities given by the operator ∆2. 1

Here, ∆ is the Laplace operator ∆ = −∇2 . It is a positive operator.

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The Fluctuation-Dissipation Theorem 10/24

The noise η is best modelled as a Gaussian with zero mean and covariance: < η(t, x)η(t0, x0) >= δ(t − t0)γ(x, x0).

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The distribution γ(x, x0) should now be determined by physical considerations. The condition that the average energy pumped into the system by noise should be balanced by the dissipation is that the covariance of the fluctuations should be proportional to the dissipation tensor: γ(x, x0) = ∆2δ(x, x0).


The Need for Regularization 11/24

Together we now can write a Fokker-Plank equation for the NavierStokes equation. Unfortunately, this is a hopelessly singular equation,because of the above distributions. We need to regularize the system. What is a regularization that preserves the Lie algebra structure? The Lie algebra of symplectic transformations can be thought of as

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the limit of the unitary Lie algebra as the rank goes to infinity. To see

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this, impose periodic boundary conditions and write in terms of Fourier

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coefficients as H = (L1L2)

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X m6=(0,0)

1 |ωm|2, 2 m

{ωm, ωn} = −

2π abmanbωm+n. L1 L2 (11)

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Using an idea of Fairlie and Zachos, we now truncate this system by imposing a discrete periodicity mod N in the Fourier index m; the structure constants must be modified to preserve peridocity and the JJ

Jacobi identity: 1 {ωm, ωn} = sin[θ(m1n2 − m2n1)]ωm+n θ

mod N ,

2π θ= . N

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This can be thought of as a ‘quantum deformation’ of the Lie algebra of symplectic transformations. This is the Lie algebra of U (N ).

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Regularized hamiltonian 13/24

The hamiltonian also has a periodic truncation 1 1 X H= |ωm|2, 2 λ(m)

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λ(m)6=0

λ(m) =

2 2 N 2π N 2π sin m1 + sin m2 . 2π N 2π N

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This hamiltonian with the above Poisson brackets describe the geodesics on U (N ) with respect to an anisotropic metric. We are writing it in a basis in which the metric is diagonal.


Now we can write the regularized Navier-Stokes with random sources. We will not have the problem of ‘closure’ that plagues many approaches to turbulence because our regularization perserves the symmetries (the 14/24

Lie algebra structure). dω m = rmn∂pH − Dmn∂nH + η n dt

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where the Poisson tensor is given in terms of the structure constants of the Lie algebra rmn = cmn p ωp . The dissipation tensor is diagonal in our basis and has the square of the eigenvalues of the discrete Laplacian:

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Dmn = νδ mnλ(m)2.Also,

J < η mη n >= Qmn for some positive covariance matrix Q.

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The Regularized Fokker-Plank Equation 15/24

The Fokker-Plank equation is now ∂W = ∂m (rmn∂mH W + Dmn∂nH W + Qmn∂mW ) . ∂t

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To have an equilibrium solution of the form W ∼ e−βH , we need to postulate a relation between the dissipation and fluctuation tensors:

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Dmn = βQmn. It is not easy to show that the limit N → ∞ exists:

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about as hard as constructing a quantum field theory. Assuming the limit

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exists we can make some predictions about two dimensional turbulence.

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Entropy of Incompressible Flow 16/24

The constants

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Qm =

R

ω m(x)d2x define some infinite dimensional

surface in the space of all functions on the plane. The micro-canonical entropy (a la Boltzmann) of this system will be the log of the volume of this surface. How to define this volume of an infinite dimensional manifold? We can compute it in the regularization and take the limit. It is P ˆ = m ωmU (m) where convenient to regard ω as a matrix by defining ω 2πi

U (m) = U1m1 U2m2 with the defining relations U1U2 = e N U2U1. In this basis, the Lie bracket of vorticity is just the usual matrix commutator. 2

Savitri V. Iyer and S.G. Rajeev Mod.Phys.Lett.A17:1539-1550,2002[ physics/0206083]


Then the regularized constants of motion are Qk =

1 N

ˆ k . The tr ω

set of herimtean matrices with a fixed value of these constants has a Q finite volume,known from random matrix theory: k 1.

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A Quantum Gas as a Regularization 23/24

If N = 1, this hamiltonian has a simple meaning: it is a many body system of n particles moving on the configuration space M subject to a pairwise potential V (φ, φ0). If we imagine now that each particle has a wavefunction valued in the space of N × N matrices and consider only states invariant under U (N ), we get exactly the above hamiltonian. We

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are not taking the limit as N → ∞: all values of N > 1 should be in

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the same universality class.

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The interesting case is when M is of positive curvature and finite volume. Then as we let n → ∞ we will approach a system of infinite density if we hold the volume fixed. There are logarithmic divergences 24/24

and the metric needs to renormalized (flattened out) to get a sensible limit. This gives an interesting new way of simulating a quantum field theory as well...
